A lower Bound for the Area of Plateau Foams

Vicent Gimeno*, Steen Markvorsen, José M. Sotoca

*Corresponding author for this work

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Abstract

Real foams can be viewed as a geometrically well-organized dispersions of more or less spherical bubbles in a liquid. When the foam is so drained that the liquid content significantly decreases, the bubbles become polyhedral-like and the foam can be viewed now as a network of thin liquid films intersecting each other at the Plateau borders according to the celebrated Plateau's laws. In this paper we estimate from below the surface area of a spherically bounded piece of a foam. Our main tool is a new version of the divergence theorem which is adapted to the specific geometry of a foam with special attention to its classical Plateau singularities. As a benchmark application of our results we obtain lower bounds for the fundamental cell of a Kelvin foam, lower bounds for the so-called cost function, and for the difference of the pressures appearing in minimal periodic foams. Moreover, we provide an algorithm whose input is a set of isolated points in space and whose output is the best lower bound estimate for the area of a foam that contains the given set as its vertex set.
Original languageEnglish
Article number96
JournalJournal of Inequalities and Applications
Volume2020
Number of pages23
ISSN1025-5834
DOIs
Publication statusPublished - 2020

Keywords

  • Foams
  • Bubbles
  • Density
  • Pressure
  • Comparison geometry

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